QIT mass spectrometers play a central role in the success of mass spectrometric methods for ionized molecules, macromolecules and biomolecules. Generally, a conventional QIT mass spectrometer includes a quadrupole ion trap (QIT) composed of a hyperbolic ring electrode and two hyperbolic end-cap electrodes to confine ionized particles therein. The ring electrode is fed with a main radio frequency (RF) waveform, and the two end-cap electrodes are fed with an auxiliary waveform, thereby trapping the ionized particles.
In a conventional method for mass spectrometry, the RF field is held at a constant frequency, and thus around a center of the QIT, the motion of trapped ionized particles approximately obeys the Mathieu equation both radially and axially. In practice, since buffer gas cooling may be used to slow down the motion of ionized particles for better motion control, a damping correction due to buffer-gas cooling can be added to the Mathieu equation as shown in Equation (1) below.
                                                                                          d                  2                                ⁢                u                                            d                ⁢                                                                  ⁢                                  t                  2                                                      +                                          (                                  a                  -                                      2                    ⁢                                          q                      ·                                              cos                        ⁡                                                  (                                                      2                            ⁢                                                          ξ                              ⁡                                                              (                                t                                )                                                                                                              )                                                                                                                    )                            ⁢              u                        +                          γ              ⁢                                                d                  ⁢                                                                          ⁢                  u                                                  d                  ⁢                                                                          ⁢                  t                                                              =          0                ;                              where            ⁢                                                  ⁢            2            ⁢                          ξ              ⁡                              (                t                )                                              =                                    ∫              t                        ⁢                                          Ω                ⁡                                  (                  x                  )                                            ⁢              d              ⁢                                                          ⁢              x                                      ;                  q          =                                    q              z                        =                                          8                ⁢                                                                  ⁢                eV                                                              m                  /                                      z                    ⁡                                          (                                                                        r                          0                          2                                                +                                                  2                          ⁢                                                      z                            0                            2                                                                                              )                                                                      ⁢                                  Ω                  2                                                                    ;                  a          =                                    a              z                        =                                                            -                  16                                ⁢                eU                                                              m                  /                                      z                    ⁡                                          (                                                                        r                          0                          2                                                +                                                  2                          ⁢                                                      z                            0                            2                                                                                              )                                                                      ⁢                                  Ω                  2                                                                    ;                              q            z                    =                                    -              2                        ⁢                          q              r                                      ;                              a            z                    =                                    -              2                        ⁢                          a              r                                      ;                                            2              ⁢                              ω                ~                                      Ω                    =                      β            ⁡                          (                              q                ,                                  a                  _                                            )                                      ;                              and            ⁢                                                  ⁢                          a              _                                =                      a            -                                          κ                ⁡                                  (                                      γ                    ,                    Ω                                    )                                            2                                      ;                            (        1        )            and the symbols used above are defined as follows:
u: r or z, the former and the latter respectively representing displacements of sample ion motion in radial direction and z-direction;
r0: inner size of the QIT in radial direction;
z0: inner size of the QIT in z-direction;
2ξ: phase of the main RF waveform;
e: elementary charge;
m/z: mass-to-charge ratio of the ionized/charged particle (m: mass, z: charge);
V: amplitude of main RF waveform;
Ω: frequency of main RF waveform (angular);
U: DC offset of main RF waveform;
β: a function of q and a, noting that β/2 is a ratio of a frequency of secular motion of the ionized/charged particle to a frequency of the main RF waveform;
γ: damping constant due to gas collision;
κ: damping coefficient, which is related to γ;
{tilde over (ω)}: secular frequency (angular); and
                              u          =                                                    ∑                                  -                  ∞                                ∞                            ⁢                                                          ⁢                              u                n                                      =                                          ∑                                  -                  ∞                                ∞                            ⁢                                                          ⁢                                                                    C                    n                                    ·                  exp                                ⁢                                                                  ⁢                                  (                                                            i                      ·                                              (                                                                              -                            β                                                    +                                                      2                            ⁢                            n                                                                          )                                                              ⁢                    ξ                                    )                                                                    ,                            (        1.1        )            where Cn is a coefficient denoted for nth component of ion displacement.
A closed-orbit solution (Equation (1.1)) is thus multi-periodic (composed of a period of RF field, and a period of secular motion of ion), as depicted in a stable region by the q-a diagram shown in FIG. 1. When the main RF waveform is applied in a manner as to let values of q and a fall outside the stable region, motions of the charged particles may become instable and the charged particles may be ejected out of the quadrupole ion trap. For a small amount (less than a few hundred in number) of molecular sample ions (or less than a few hundred in elementary charges) in a QIT having a stable gas flow and a slowly-ramping RF amplitude, highly sensitive mass spectrometry (MS) with a resolution over one thousand can be achieved.
When the number of molecular sample ions is increased to the thousands (such as MALDI sample ions) and the charges of the molecular sample ions are increased (such as LIAD sample ions), the sample ion-ion interactions become non-negligible, intervene in the kinetics of buffer gas cooling, and induce additional randomness upon the spectrometric path for mass discrimination within the q-a diagram. Therefore, the spectral outcome may become rather scattered, besides having substantial deviations, much away from what has been considered ideal according to the Mathieu equation.
In order to avoid scattered mass spectral outcome with substantial deviations from ideal location of mass spectral peaks, the ion-ion interaction is re-formulated into the collisional damping as being represented as a stochastic cut-off. Then, the dynamical equation of trapped ions uses the definite phase of main RF field as an independent variable, and thus an inherent dispersion can be explicitly discerned. Thus, to maintain the interpretation of simple Mathieu equation, advanced modulation process and detection technique were developed for such “ion cloud” spectrometry.
Either with molecular ions injected into the QIT or with molecules directly ionized inside the QIT, from the q, a values of ion motion, instability of molecular ions can be discerned within a few RF cycles. Constant-frequency trapping coincides ion's dynamics (i.e., the dynamical equation) wish the Mathieu equation, and is to use amplitude ramping in the main RF amplitude for linear mass spectrometry. Efficient cooling through gas collisions makes high-resolution mass spectrometry feasible, as if the kinetic deviation in accuracy can be calibrated or neglected. However, adjustable magnification for the amplitude of the main RF waveform has physical limitations, so mass scan is limited within a relatively small range.